Analysing the mechanical performance and growth adaptation of Norway spruce using a non-linear finite-element model and experimental data

doi:10.1093/jxb/ern116

RESEARCH PAPER

Analysing the mechanical performance and growth

adaptation of Norway spruce using a non-linear

finite-element model and experimental data

T. Lundstro¨m1,2,*, T. Jonas1and A. Volkwein1 1

WSL, Swiss Federal Institute for Snow and Avalanche Research SLF, 7260 Davos Dorf, Switzerland 2

Laboratory of Dendrogeomorphology, University of Fribourg, 1700 Fribourg, Switzerland Received 14 December 2007; Revised 26 March 2008; Accepted 31 March 2008

Abstract

Thirteen Norway spruce [Picea abies (L.) Karst.] trees of different size, age, and social status, and grown under varying conditions, were investigated to see how they react to complex natural static loading under summer and winter conditions, and how they have adapted their growth to such combinations of load and tree state. For this purpose a non-linear finite-element model and an extensive experimental data set were used, as well as a new formulation describing the degree to which the exploitation of the bending stress capacity is uniform. The three main findings were: material and geometric non-linearities play important roles when analysing tree deflections and critical loads; the strengths of the stem and the anchorage mutually adapt to the local wind acting on the tree crown in the forest canopy; and the radial stem growth follows a mechanically high-performance path be-cause it adapts to prevailing as well as acute seasonal combinations of the tree state (e.g. frozen or unfrozen stem and anchorage) and load (e.g. wind and vertical and lateral snow pressure). Young trees appeared to adapt to such combinations in a more differentiated way than older trees. In conclusion, the mechanical performance of the Norway spruce studied was mostly very high, indicating that their overall growth had been clearly influenced by the external site- and tree-specific mechanical stress.

Key words: Climate, critical load, mechanical optimisation, model performance and errors, stem taper, structural behaviour, thigmomorphogenesis.

Introduction

When a tree deflects, there is a change in the way loads,

such as wind and snow, apply upon the tree. This

phenomenon, which is called geometric non-linearity,

is considered significant for a tree if the top deflection

relative to height is larger than 20%. Until now,

geometric non-linearity has rarely been fully considered

in the mechanical analysis of flexible plants (Yang

et al.,

2005). In addition, trees also exhibit material

non-linearity because they are composed of natural materials.

This means that their mechanical properties depend on

the magnitude of the load. Typically, the rotation of the

root–soil system depends, in a non-linear way, on the

applied

turning

moment

of

the

root–soil

system

(Lundstro¨m

et al., 2007c), just as the stem deflection

due to bending depends on the applied bending stress

(Lundstro¨m

et al., 2008). These material non-linearities

have never, to the author’s knowledge, been considered

in the analysis of tree mechanics. Consequently, it is not

known which role they play in how a tree reacts to

external load or under what conditions they may be

mechanically advantageous for the tree, as they

contrib-ute to its flexibility.

Stresses in the tree stem and anchorage result from

combinations of wind and snow load, and the overhanging

weight of the leaning tree. The magnitude and frequency

of such loads depend on the tree and stand characteristics,

climate, and season. A sufficiently heavy load will cause

the stem or the tree anchorage to fail. The type of failure

depends on the stem and anchorage strengths, and on

where and in which direction the load is applied. The

magnitude of the failure load will, in addition, depend on

how well the tree, including the crown, stem, and

* To whom correspondence should be addressed. E-mail: [email protected]

ª The Author [2008]. Published by Oxford University Press [on behalf of the Society for Experimental Biology]. All rights reserved. For Permissions, please e-mail: [email protected]

anchorage, has adapted its growth to the particular load

combination (Telewski, 1995; Nicoll and Ray, 1996; Di

Iorio

et al., 2005). The most common failure mode of

Norway spruce [Picea abies (L.) Karst.] is uprooting, but

stem breakage may occur if the soil is frozen and if the

crown is loaded with snow (Peltola

et al., 1997, 2000). It

is not clear to what extent a tree mutually adapts its stem

and anchorage strengths to resist various types and

combinations of natural loads.

Thigmomorphogenesis is the response of plants to

mechanical stress (Jaffe, 1973; Telewski, 2006), with

respect to their morphology and material properties

(Chiatante

et al., 2002). So far, most studies of

thigmo-morphogenesis have focused on isolated parts of the plant

rather than on the plant as a whole (Moulia

et al., 2006).

Clearly, the spatial and temporal focus of such studies will

affect how much detail they give in assessing and

analysing the stress and growth histories of plants.

Wind-induced stem-bending stress has been the topic

of several studies (Milne, 1991; Morgan and Cannell,

1994; Wood, 1995; Dean

et al., 2002), where stress

uniformity along the stem has been attributed to its

taper. In addition to this geometric adaptation, the stem

also adapts anatomically to mechanical stress through

differential radial and apical growth (Meng

et al., 2006).

As the tree grows larger, this normally leads to stronger

and stiffer wood forms along the lower, outer part of the

stem (Ylinen, 1952; Niemz, 1993; Lundstro¨m

et al.,

2008), where the largest bending stresses due to wind

are expected. Studying the thigmomorphogenesis of

the stem, therefore, seems to require including both the

taper and the material properties of the stem in the

analysis.

This paper investigates how Norway spruce reacts to

combinations of natural loading. For this purpose,

a non-linear numerical tree model is provided with

extensive experimental data. These data are first used to

test the model performance. Then, on the basis of the

model results, the stem thigmomorphogenesis is

ana-lysed with a novel method that considers both the taper

and the material properties of the stem. The focus is on

the mechanics of the tree as a whole, and the properties

of growth were used to parameterize its mechanics. The

findings are used to test three hypotheses: (i) a

geo-metric and material non-linear analysis is required to

reproduce

in situ stem deflections and analyse critical

loads; (ii) the strengths of the stem and the tree

anchorage mutually adapt to the most frequent

combina-tion of seasonal tree-state and load; and (iii) the

thigmomorphogenesis of the stem is the result of the

prevailing combination of seasonal tree-state and load.

Exploring these hypotheses should improve researchers’

general understanding of how Norway spruce grows and

thus contribute to the informed management of Norway

spruce forests.

Materials and methods

Trees and sites

The selection of the 13 Norway spruces [Picea abies (L.) Karst.] analysed was intended to cover not only a range of frequent tree sizes and of typical growth conditions in the Alps (Schweingruber, 1996), but also a spectrum of different tree ages, social states, sites, and local stand densities. By taking into account these factors, it would be possible to analyse the tree growth response generally and spatiotemporally according to the mechanical loading. The site and tree characteristics relevant to this study are listed in Table 1. These, as well as all experimental data used in this study, have been obtained from plots used in winching tests (Kalberer, 2007; Lundstro¨met al., 2007b, c). The standing tree is referred to with an (x, y, z)-coordinate system, with its origin (0, 0, 0) at the stem base, z¼H at the top of the tree, and x in the direction of the horizontal stem deflection.r is the radial distance from the pith, DBH the stem diameter over bark at breast height (z¼1.3 m), and B the bark thickness. The symbols used in this paper are summarized in Table A1 in the Appendix.

Table 1. Site and tree characteristics of the 13 spruces analysed

Plot, local delimitation of the forest stand;DBH, stem diameter at breast height (1.3 m); H, tree height; LC, crown length;zCc, height of the crown’s area centre; AGE, cambial age; Social status (according to Dobbertinet al., 1997) in relation to neighbouring trees.

Plot no. Tree no. Elevation (m asl) Exposition Slope (
) Stand density (trees ha
1) DBH (cm) H (m) LC/H (
) zCc/H (
) AGE atz¼1.3 m (years) Social status 1 11 1620 NE 35 1200 15 16 0.88 0.56 64 Dominated 2 12 1620 NE 35 800 15 15 0.86 0.53 47 Co-dominant 3 13 1780 ESE 31 500 22 21 0.76 0.57 103 Dominated 3 14 1780 ESE 31 500 35 26 0.80 0.56 118 Co-dominant 4 15 1650 NNW 33 500 44 32 0.57 0.68 177 Dominant 3 16 1780 ESE 31 500 52 33 0.87 0.51 267 Dominant 3 17 1780 ESE 31 500 58 35 0.82 0.52 271 Superior 5 21 460 NNW 4 350 19 17 0.82 0.56 28 Co-dominant 5 22 460 NNW 4 350 29 26 0.76 0.56 43 Co-dominant 6 23 620 NNE 5 250 40 36 0.44 0.77 85 Co-dominant 6 24 620 NNE 5 250 48 36 0.36 0.80 84 Dominant 7 25 620 NNE 5 120 65 40 0.60 0.67 83 Dominant 8 26 620 NNE 5 80 73 38 0.66 0.65 81 Dominant

The trees 11–17 (Table 1) had a mean B(z¼1.3 m)/(DBH/2) of 7%. They grew at a high elevation (HE) on four closely situated plots (at about 46
46# N, 9
49# W) south of Davos, Switzerland. At these locations, the B-horizon (0.1 m<depth<0.4 m) is a dystric Cambisol. The 5:50:95% percentiles of daily temperature observed over the last 20 years are –8:+4:+15
C and of 10-min-wind 0:2:6 m s
1. The annual maximum 10-min-wind is 15 m s
1and the annual precipitation 1100 mm. Almost half the precipitation falls as snow and there is snow cover for 6 months a year (MeteoSwiss, 2007. Climate database of the Swiss National Weather Service, http:// www.meteoschweiz.ch/web/en/services/data_portal.html). The soil-frost depth generally increases from November to culminate with a mean:max depth of 0.25:0.75 m beneath the crown in April (Stadleret al., 1998; MeteoSwiss, 2007, http://www.meteoschweiz. ch/web/en/services/data_portal.html).

The trees 21–26 had a meanB(z¼1.3 m)/(DBH/2) of 6%. They grew at a low elevation (LE) on four plots close to Zurich, Switzerland, with plots 5–6 at 47
14# N, 8
53# W and plots 7–8 at 47
22# N, 8
28# W. At these locations, the B-horizon (0.4 m<depth<1.0 m) is a dystric Cambisol with presence of Luvisol. Here, the 5:50:95% percentiles of daily temperature are –2:+10:+21
C and of 10-min-wind 0:2:6 m s
1. The annual maximum 10-min-wind is 22 m s
1, and the annual precipitation 1150 mm. Almost 10% of the precipitation falls as snow and there is snow cover for about 33 d a year (MeteoSwiss, 2007, http://www.meteoschweiz.ch/ web/en/services/data_portal.html). The maximum soil-frost depth beneath the crown is less than 0.10 m (Stadler et al., 1998; MeteoSwiss, 2007, http://www.meteoschweiz.ch/web/en/services/ data_portal.html).

Attribution of mechanical properties to the tree model

The tree is divided into a sufficient number of equally long elements for which the mechanical properties are specified. In this study 100 elements are used. These elements have the idealized shape of circular cylinders, because the stem diameter on bark (D) differed little in thex- and y-directions (2% on average). Each element is attributed a stem diameter under barkDub(z), according to Table 2

using the following measured data: D in the x- and y-directions every per cent ofH up to z/H¼0.20, and then every metre up to H; B in the x- and y-directions at z/H¼0.05 and at least two additional stem heights (in stem discs). Each element is further attributed a stem weight and a crown weight and diameter, which are polynomials ofz based on measurements every metre of tree height.

Here, the crown diameter is the geometric mean of the extensions in thex- and y-directions. The stem-section elements are finally given the following deformation properties: a bending stress as a non-linear function of bending strain and stem height r(e,z) (Equation 1, an approximation) and a shear stress as a non-linear function of shear strain s(c) (Equation 2, a simplification):

rðe; zÞ ¼
_rðeÞ rmax  3
_r maxðzÞ rmaxð0:05HÞ  3rmaxð0:05HÞ ð1Þ sðc; zÞ ¼ sðcÞ ¼
_rðeÞ rmax  3smax ð2Þ

where rmax is the bending strength of the stem cross-section

(sometimes called the modulus of rupture) and smax is the shear

strength of the stem section (Table 3). The first two factors in Equation 1 are normalized mean (roof sign) curves of r(e) and of rmax(z),

respectively (Lundstro¨met al., 2007a, 2008). The stem base is flexibly clamped to the ground. The lower end of the lowest element (atz¼0) is therefore attributed a resistive rotational momentM0as a non-linear function of the rotational angle / (Equations 3 and 4):

M0ð/Þ ¼ M0 max3Mn0ð/nÞ ð3Þ With M0n¼ M0 M0 max ;/n¼ / /ðM0 maxÞ ð4Þ where M0

n is the mean Mn0ð/nÞ-curve for spruces at the two elevations (Lundstro¨met al., 2007c), M0

max is the maximum ofM

0

(sometimes called the anchorage strength), and /ðM0

maxÞ is the / at M0

max (Table 3). The idealized M

0

-shape, resulting from M0 n, was used instead of the measured M0-shape in order to simplify and systemize the modelling. This approximation was tested to make sure it had no significant effect on the results.

All the parameters in Table 3 are measured values (winching experiments), except for rmax and smax. The cross-sectional rmax

was calculated on the basis of the annual ring width RW and the knottinessQ between the pith (radius r¼0) and the bark (r¼Dub/2) as

variables in statistical-mechanical models (Lundstro¨m et al., 2007a, 2008).RW was scanned and measured digitally (WOODSCAN 4.5, Freiburg University, Germany) in stem discs, in eight azimuthal directions as a function ofr, and then averaged at the same age. Q was measured manually in digital photos of the discs and averaged accordingly. Data onRW(r) and Q(r) were obtained from at least four discs cut from the stem betweenz¼0 and z¼H, in addition to the disc atz¼0.05H. The values of rmax, calculated on the basis of theRW(r)

and Q(r) in these additional discs, differed little from the rmax

(z)-values obtained with Equation 1. The latter rmax(z)-curves were

therefore used throughout the analysis. The cross-sectional smaxwas

estimated (Lundstro¨met al., 2008) on the basis of the mean RW of the inner 75% radial part of the stem cross-section (RWi).RWiwas

found to vary little with stem height, so smax(z) was set as a

constant¼ smax(z¼0.05H).

It is known that frost increases the strength and stiffness of moist wood and soil (Kollmann, 1968; Andersland and Ladanyi, 2004). Silinset al. (2000) showed that MOE(z) of fresh wood increases by a factor 1.5. Trees winched in the winter at the HE-site (SLF, unpublished data) show thatM0maxincreases by a factor2.5. These factors were adopted to account for the frozen stem and soil. Loads on the tree and combinations of tree state and snow load

In this study, intercepted snow provides the vertical load on the tree. Here, two nominal values of snow-water equivalent precipitation Table 2. Sequential steps in the parameterization of the stem

diameter as a function of stem height

D, stem diameter over bark; Dub, stem diameter under bark;B, bark thickness. Step 4 is not used by the tree model, only to analyse stem morphology.

Step Description

1 Polynomial fit ofDx(z) and of Dy(z) with a polynomial degree so that the fitting error is <1% for the 2/3 lower stem and <5% for the 1/3 upper stem (the resulting degree was >16). 2 Calculation ofBx(z) and By(z) using a 5-degree polynomial of

B(z)/D(z) (Laasasenaho et al., 2005) scaled with the measured BxandByatz/H¼0.05. The fitting errors at the additional B(z)-measurements were all within the limits of the absolute values at the corresponding heights in step 1.

3 Calculation ofDubx(z)¼ Dx(z)–2Bx(z), Duby(z)¼ Dy(z)–2By(z), and the geometric meanDub(z)¼ (Dubx(z)Duby(z))0.5. 4 Calculation of the normalized stem diameterD05(z)¼ D(z)/

D(z/H¼0.05), in the x- and y-direction, as the geometric mean, under and over bark.

were first considered, SWE01¼20 and SWE02¼40 mm. The SWE

effectively intercepted by the crown area projected on the ground was then calculated as SWE¼ SWE03p/43atan(qcv/10), where qcv

is the vertical crown density¼ total crown mass/projected crown area on the ground, and 10 kg m
2 is the reference qcv. This

expression accounts for the fact that a denser crown has a better capacity to intercept snow (Strobel, 1978). The total snow loads on the crown, resulting from SWE01and SWE02, are denoted SL1 and

SL2, respectively. They were distributed height-wise in proportion to the mass per height-section of the crown. This meant the top of the crown was attributed relatively more snow than its base because the crown density increased with height. SL1 was estimated to be the maximum amount of snow that can remain on the crown of a spruce of mean size when subjected to strong wind and SL2 the amount when there is almost no wind (Pomeroy and Gray, 1995; Bru¨ndl, 1997). The height-wise mean snow weight along the crown ranged between 6 kg m
1 and 48 kg m
1 for SL1 and between 12 kg m
1and 95 kg m
1for SL2.

The eight most significant combinations of seasonal tree state and snow load analysed are listed in Table 4, along with their probability of occurrence. Here, SL1 includes 10<SWE01< 30

mm and SL2, SWE02>30 mm.

The lateral load, resulting from, for example, wind or snow pressure, was simplified and reduced to one concentrated force, F(z). The next section describes how F(z) was applied to the tree. Tree model and model simulations

The tree model used is two-dimensional (Fig. 1). Since its single elements are one-dimensional, the forces and deformations along the stem, due to external forces, can be computed with the transfer matrix method. This method, which is based on the classical finite-element analysis of structural mechanics (Cook et al., 2002), is described in detail in Morgan and Cannell (1987) and Ancelinet al. (2004). In addition to the formulation in Morgan and Cannell (1987), the present method considers the geometric non-linearity and the material non-linearity of the deflecting tree.

The two non-linearities are taken into account by repeatedly performing the transfer matrixes method in its linear form, in an iterative process, until the tree model achieves static equilibrium. The geometry non-linearity is considered by redefining, at each repetition and by simple trigonometric transformation, the local coordinates (x#, z#) of each element in the global coordinates (x, z) in which the external forces act. Similarly, the material non-linearity

is considered by redefining, at each iterative step, the secant-modulus of bending elasticityE(e)¼ r(e)/e (cf. Equation 1) and of shearing elasticityG(c)¼ s(c)/c (cf. Equation 2) of the elements, to fit the r(e) and the s(c) acting on the element. Here, the element deflection dx#(z#) due to the shear force (V, Fig. 1) is calculated with the mean shear stress of the cross section:

s_{ðz#Þ ¼} k

AðzÞ3Vðz#Þ ð5Þ

as s(c,z#) in Equation 2. This yields c(z#) and thus:

dx#ðz#Þ ¼ c3ðz#Þ3dL ð6Þ where k¼1.1 is a shape factor for a circular cross-section to take into account the effective shear area,A(z)¼ pDub(z)2/4 is the area

of the cross-section, and dL is the length of one stem element. What is notable here is that the shear deformation of the stem due to asymmetrical stresses along the stem (dz#/dx#) is very small compared to that due to the slip along the wood grain (dx#/dz#), and that therefore c_{dx#/dz#. The deformation along the stem is} simply calculated with the stress-strain relationship in Equation 1, with the longitudinal stem stress¼ N(z#)/A(z) in place of r and e interpreted as the longitudinal stem strain, where N is the normal force (Fig. 1). The maximum shear stress within the cross-section is computed as s(z#,r¼0) ¼ 4/33V(z#)/A(z). The material non-linear-ity of the root–soil system is considered by redefining, at each iteration, its secant-stiffness¼ M0

(/)//(Equation 3) to fit the M acting on the stem base. At the same time a small rotation is added or subtracted to the stem base, corresponding to a change in the stem base momentM0(/) to match the erroneousM at the tree top. The convergence criterion in the iteration to achieve a static equilibrium of the tree model is that the absolute valuesN, V, and M (Fig. 1) are all < 1E-06 (N and Nm) at the tree top.

On a second and a higher level of iteration, the critical lateral forceFcrit(z#) that causes tree failure is sought. Fcrit(z#) corresponds

to the lower of the forces causing uprooting, i.e.M0_{ M}0 max, and stem breakage, i.e. r(e,z#)rmax(z#). The corresponding static

equilibriums are denoted StEq(M0

max) and StEq(rmax). The initial

and estimated F(z#)-value is set as a percentage of M0

max=zðFÞ, to which a small value is then iteratively added or subtracted until the absolute values of M0
_M0

max 

=M0

max or (r(e,z#)–rmax(z#))/

rmax(z#) are less than 5&. To avoid model divergence, the E¼r/e

at rmax is used to calculate r beyond rmax. This is of no

importance for the interpretation of the results other than that Table 3. Mechanical and growth characteristics of the trees analysed

RW, annual ring width, where ‘i’ refers to the mean of the inner 75% radial part of the stem cross-section, and ‘o’ to the mean of the remaining outer radial part; qw, bulk density, rmax, bending strength, and smax, shear strength of the fresh stem. The other abbreviations are explained in the text.

Tree no. M0 max (KNm) /(M0 max) (
)

Values of the stem section atz¼0.05H

RWi(mm) RWo(mm) qw(kg m

3

) rmax(MPa) smax(MPa)

11 12 21.0 1.1 1.3 840 52.6 3.5 12 10 22.0 1.4 2.3 780 46.0 3.1 13 30 10.0 1.5 0.6 880 59.6 3.1 14 75 7.4 1.6 1.0 830 54.0 3.0 15 139 5.3 1.3 0.9 860 56.6 3.3 16 231 3.3 0.9 1.1 870 55.4 3.8 17 239 3.0 0.9 0.6 920 62.9 3.7 21 41 18.0 3.7 3.1 600 38.8 1.8 22 98 8.4 3.3 3.4 610 39.1 2.0 23 228 5.5 2.5 2.2 740 44.0 2.3 24 308 4.8 3.4 2.0 710 42.2 1.9 25 753 3.3 4.0 3.5 660 36.1 1.7 26 883 2.6 4.3 4.6 640 33.9 1.6

r(z#)>rmax(z#) indicates stem-bending failure. Figure 2

demon-strates typical model results at StEq(M0max) (Fig. 2A–D) and at StEq(rmax) (Fig. 2E–H), which are here exemplified for two trees in

an unfrozen state with no snow load.

Figure 2A–D shows tree no. 22 at StEq(M0

max). The lateral force F is applied at z#/H¼0.55 (Fig. 2A), which is close to the crown’s area centre,zCcatz/H¼0.56. F causes a total stem deflection (black

line) and a stem deflection due to shearing only (grey line). The position of the initially applied forceFinit(dashed line and arrow)

differs from that of the converged solutionFcrit(continuous line and

arrow). Note thatF is applied horizontally (in the x-direction) and at one fixz#-coordinate. For tree no. 22, uprooting occurs, i.e. the stem base momentM0(/) reachesM0

max (Fig. 2B), before the stem fails, i.e. before the stem-bending stress r(z#) reaches the stem-bending failure stress rmax(z#) (Fig. 2C), where r(z#) is the actual stress

(solid line) and rmax(z#) the failure stress (dashed line). The actual

effective secant-modulus of elasticity of the stem section,E(z#) ¼

r(z#)/e(z#) (Fig. 2D, continuous line), is, along the lower stem, lower than the modulus defined at 0.4rmax(z#), MOE(z#) (dashed

line). The difference results from the non-linear r(e,z) and means that the stem section softens while bending as r(z#) approaches rmax(z#),

i.e. it bends more easily thanMOE(z#) would predict.

Figure 2E–H shows tree no. 23 at StEq(rmax).F is applied at

z#/H¼0.85 (Fig. 2E), just above zCc at z/H¼0.77, and causes

considerable stem deflections. In this example, the stem fails, r(z#)rmax(z#) (Fig. 2G), before the tree uproots, i.e. before M

0

(/) reaches M0max (Fig. 2F). The softening of the stem section while

bending (Fig. 2H) is more extensive than in Fig. 2D.

The tree model was run for Fcrit at the relative tree heights

z#/H¼0.10, 0.15, 0.20, ., 0.95, in order to determine the dependency of tree reactions on the height of the force applicationz#(Fcrit). Here,

all combinations of tree state and snow load (Table 4) were included to determine the dependency of tree reactions on the season.

To evaluate the model performance, the modelled and the measured stem deflections were compared at StEq(Mmax0 ), withFcrit

atz#/H¼0.20, for all trees in their normal state (Table 4). Critical wind speed

The critical forcesFcrit(z#/H¼0.05, 0.10, ., 0.95) were interpolated

every per cent ofH up to 95%. Fcritat the height of the crown’s

area centre,zCc(cf. Table 1), was used to calculate the equivalent

critical speed of steady wind,ucrit, with Equation 7:

F¼ CD3 q3u2

2 3A0; CD¼ a þ ð1
aÞ3e

au _ð7Þ

where CD is the crown drag coefficient, A0 is the horizontal

projection of the undeformed crown, q is the density of air¼1.17 kg m
3, a¼–0.09, and a¼0.18 (Lundstro¨m et al., 2007b). Snow on the crown is accounted for with a¼a/(1+atan(SWE/40), which is a hypothesis. Frozen branches are accounted for with a¼a2/3, as frozen wood is 50% stiffer (Silinset al., 2000), which is applied when the stem is frozen (cf. Table 4). The two effects on a are multiplicative.

Stem thigmomorphogenesis

To analyse the degree to which the stem adapts its growth to mechanical stress (thigmomorphogenesis), the r(z#) calculated by the model is compared with rmax(z#). For this, a new formulation

(Equation 8) was developed: gðz19Þ ¼ SE z91 z9¼0 rðz9Þ rmaxðz9Þ 3rmaxðz9Þ rðz9Þ !!!
1 ð8Þ Table 4. Combinations of seasonal tree state and snow load in the model calculations

HE, high elevation site; LE, low-elevation site.Dubis the stem diameter under bark. SL and SWE are explained in the text.

Abbreviation Description Annual mean occurrencea(d)

Tree state Snow load HE LE

0 Normal 260 360

0 _SL1 Normal+snow load (SWE01) 45 5

0 _SL2 Normal+snow load (SWE02) 1 <0.1

FrSt Frozen stem (depth >0.1Dub) 40b 0

FrSt _SL1 Frozen stem+snow load (SWE01) 18b 0

FrSo Frozen soil (depth >20 cm) 10b ₀

FrStSo Frozen stem and soil 7b 0

FrStSo _SL1 Frozen stem and soil+snow load (SWE01) 2b 0

a_{Variations of up to ten times the mean occur for the abnormal combinations.} b_{Estimation based on Stadleret al. (1998) and Zweifel and Hasler (2000).}

Fig. 1.The tree (grey dashed lines) and the model tree (heavy black lines) with schematic finite height segments of the tree. Each element is a one-sided clamped beam. For clarity, only one element of the deflected tree is shown with the moment (M) and forces (N, V) acting at its upper node. (x, z) refer to the global coordinate system and (x#, z#) to the local system of the element:z# ranges from 0 to H along the stem centre andx# is the sideway deflection of each element. M0_{(/) is the} resistive moment of the root–soil system as a function of the rotational angle at the stem base.

where g is the degree of uniform exploitation of bending stress capacity for the stem section between the stem base (z#¼0) and the height of force application ðz# ¼ z1#Þ, SE is the standard error, rmax(z#) is the bending strength (Table 3, Equation 1), and z1# is the (local) height coordinate of force application. Here, r(z#) corre-sponds to the stem-bending stress at StEq(M0

max) or StEq(rmax). In

principle, it would be more correct to use the most frequent stem stress. However, the most frequent values ofF(z#/H¼0.05, 0.10, ., 0.95), which correspond to the particular combinations of tree state and snow load, area priori unknown. When the r(z#) due to the mean wind speed was calculated and then magnified so that the shape of the r(z#)-curve could be compared with that of r(z#) due to Fcrit(zCc), it became apparent that the two r(z#)-shapes were

almost identical (see the example given in the Results). The stem thigmomorphogenesis was therefore analysed using r(z#) at

StEq(M0

max) or StEq(rmax). It should further be noted that g is

mathematically independent of the magnitude of rmax and r(z#)/

rmax(z#), and that g measures the uniformity of exploitation of

rmax(z#) and not the magnitude of its exploitation.

Results

Model performance and potential sources of error when modelling stem deflections

The relative difference between modelled and measured

horizontal tree-top deflection ranged from 1% (4 cm, tree

26) to 9% (45 cm, tree no. 22), averaging 3%. Since the

exact rotation of the root–soil system was specified, these

Fig. 2. Example of results with model convergence at StEq(M0

discrepancies were due to errors in the parameterized

flexibility of the stem only. Stem deflection could be

modelled with very small errors up to the crown base.

From here, small, local variations in flexibility accumulated

or mutually balanced along the stem up to the tree top, with

no sign of systematic errors. The errors in tree-top

deflection were considered small and irrelevant compared

to the other potential sources of errors outlined below, and

the model performance was therefore fully acceptable.

Neglecting the geometric non-linearity means ignoring

the additional moment resulting from the

x-wise deflection

of tree and snow weight, and therefore overestimating

F

(z). It also means ignoring the negative z-wise

deflection of the leaning tree, which results in a shorter

lever arm between the force application and the stem base,

and therefore an underestimation of

F

(z). The net effect

is an overestimation of

F

(z) if a geometric linear

analysis is made. The overestimation for the trees with no

snow load ranged from 5% (large tree) to 25% (small

tree), averaging 14%. With snow on the crown, these

values doubled and even tripled. In fact, some trees with

snow-loaded crowns would have required almost no

lateral force for tree failure to occur. This so-called

unstable equilibrium would have been missed if the

geometric non-linearity had been ignored.

Not accounting for the material non-linearity of the

stem, i.e. using the stress-independent stem-bending

elasticity

MOE(z) (the secant-modulus at 0.4r

), leads

to an underestimation of the stem-top deflection by

between 0% and 19%, and on average by 8%. The

relative softening of the stem in bending was typically

a few per cent and reached locally up to 42%. If the

non-linear resistive moment of the root–soil system is not

taken into account, and instead a constant stiffness (the

secant-stiffness at 0.4M

) is assumed in analogy to the

stem section, then the tree-top deflection will be

under-estimated by a factor of between 5 and 7, and on average

6. If the tree deflections are underestimated, then

F

(z)

will be overestimated. The net effects for material

non-linearity are similar to those with geometric non-non-linearity.

Hence, for trees with no snow load, the accumulated net

effect of neglecting both material and geometric

non-linearity in the analysis meant that

F

(z) was

over-estimated by between 7% (large tree) and 35% (small

tree), and on average by 20%.

Tree reactions and critical loads: generalities

Figure 3 shows characteristic trends of the critical failure

load

F

and of tree reactions according to tree height. Up

Fig. 3. (A) Values of critical lateral forceFcritas a function of the relative stem height of force application,z#(F)/H, for tree no. 15 in the unfrozen state with no snow load. Stem deflections (B), shear stress (s) in the centre of the stem (C), and stem-bending stress (r) (D) due toFcritapplied at z#(F)/H¼5, 10, ., 95%. The dotted line indicates the r due to the mean wind (1.7 m s
1) magnified r-wise to coincide with the r(z#) at StEq(M0_max). The dot-dash line (rmax) is the stem-bending strength. Continuous lines refer to StEq(Mmax0 ) and dashed lines to StEq(rmax). Heavy black lines (B–D) refer to the best 5%-multiple approximation for the relative height of the crown’s area centre (zCc). HerezCc/H¼68%70%.

to a certain height of force application (here

z#(F

)/

H¼87%), the tree uproots, i.e. StEq(M

max

) applies

(continuous lines).

F

displays here a function of type

F

;z#

. Above this height, the tree fails in the stem,

i.e. StEq(r

) applies (dashed lines), and

F

(

z#)

decreases more strongly than indicated by

F

;z#

.

The height at which this switch occurs depends on the

degree of exploitation of bending stress capacity, r(

z#)/

r

(

z#). The probability of shearing failure of the stem

is highest when

F

(

z#) applies at the stem base (cf. Fig.

3C; Table 3). If

F

applies above

z#/H¼0.05, the stem

is more prone to bending failure than to shearing failure.

Further, the r(

z#)-curve due to F

and the r(

z#)-curve

due to the mean wind have similar shapes (Fig. 3D). The

r(z#)-curve is the most similar in shape to the r

(

z#)-curve if

F

applies just above the height of the crown’s

area centre,

z

. Compared with the unfrozen trees,

r(z#)/r

(

z#) was, as expected, lower if the stem was

frozen, higher if both the stem and the soil were frozen,

and much higher if only the soil was frozen.

Further-more, as expected, stem deflections were relatively

larger for small than for big trees. For example, when

F

was applied at

z

, the

x- and z-wise tree-top

deflections relative to

H were at least 9% and 1%,

respectively (tree 17, FrSt) and at most 63% and 30%

(tree 11, 0_SL1), for all trees and combinations of tree

state and snow load.

Stem-bending stress in relation to tree state and snow load

The spruces growing at the HE-site showed a low

exploitation

of

their

stem-bending

capacity,

_r(z#)/

r

(

z#), when their M

and the resulting uprooting

was reached, i.e. at StEq(M

max

). This is due to their

relatively weak anchorage and strong stems.

Conse-quently, the stem bending strength and the resulting stem

failure, i.e. StEq(r

), was reached only if

F was applied

high up the stem. r(

z#)/r

(

z#) at StEq(M

) was

especially low for large trees (Fig. 4), meaning they were

less likely to fail in the stem than small trees (Fig. 5). In

fact, the largest trees, nos 16 and 17, were capable of

achieving StEq(M

) in all combinations of tree state and

snow load even if

F was applied above z

. In contrast,

the small HE-spruces in frozen soil failed in the stem if

F

was applied at

z

(e.g. no. 11 in Fig. 5E–G). As

a consequence of the relatively weaker stem, the small

HE-trees were generally more sensitive to snow load than

the large trees (cf. Figs 4B, 5B). In fact,

F

was very low

for the trees nos 11–13 in the combination 0_SL2,

independent of the height of application. This

phenome-non of unstable equilibrium resulted in stem failure in the

upper part of the crown at 0.70 <

z#/H<0.95.

The spruces growing at the LE-site differed from those

at the HE-site in that r(

z#)/r

(

z#) at StEq(M

) was

higher, meaning stem failure was more probable (cf. Fig. 6

Fig. 4. Stem-bending stress (r) due toFcritfor tree no. 16 in four combinations of tree state and snow load (cf. Table 4). Symbols are explained in Fig. 3.

with Figs 4A and 5A). A further contrast with the HE-site

was that the large LE-trees were more prone to stem

failure than the small trees, with overall higher r(

z#)/

r

(

z#) at StEq(M

) (cf. Fig. 4B with Fig. 5B).

Logically, in the tree-state and snow-load combination

0_SL1, the snow load caused overall higher r(

z#)/

r

(

z#), just as it did with the HE-spruces. In the

combination 0_SL2, the LE-spruces nos 22–25 failed in

the stem at 0.50 <

z#/H<0.95, with a very small F

required at any height (unstable equilibrium), whereas nos

21 and 26 showed simply generally higher r(

z#)/r

(

z#)

at

F

compared to 0_SL1.

Adaptation of stem and anchorage strengths to site-specific climate and loads

The HE-spruces displayed a well-balanced relation

be-tween the stem and the anchorage strengths when they

were subjected to the combinations of tree-state and load

application that can be expected during the year (Table 5).

Thus, the simultaneous achievement of

M

¼M

max

and

r(z#) ¼ r

(

z#), i.e. StEq(M

and r

) was, for the

unfrozen trees, often reached with

F

at about

z# ¼ 0.80H

(i.e. 0.2–0.3H above z

). This corresponds to the effective

wind loading (cf. Discussion). If the soil and the stem were

frozen, then

z#(F

) at StEq(M

and r

) was lower.

With the smaller HE-trees, it shifted down towards the stem

base, where, for example, pressure from gliding snow acts.

In

the

combination

0_SL1,

the

HE-trees

achieved

StEq(M

max

and r

) if

F

was applied at the same

z#/H-coordinate as the combination 0 or just a few per cent

lower. In the combination 0_SL2 (nos 14–17), the

corresponding downward shift was similar or just a few

per cent more.

Most LE-trees (nos 21, 22, 24, and 26) achieved

StEq(M

and r

) with

F

acting close to

z

.

However, this balance was less evident than for the

Fig. 6. Stem-bending stress (r) due toFcritin the normal tree-state/snow-load combination (0), for the trees nos 21–26 (A–F) withzCcclose toz#/ H¼55, 55, 75, 80, 65, and 65% (heavier lines). StEq(M0

maxand rmax) occurs withFcritapplied at aboutz#/H¼77%, 66%, 29%, 56%, 25%, and 41% (A–F), where the continuous line shifts to a dashed line. Symbols are further explained in Fig. 3.

HE-trees, because the LE-trees sometimes lacked stem

resistance. This was especially obvious with the LE-trees

nos 23 and 25, which achieved StEq(M

and r

) with

F

acting at the lower part of the stem. In the combinations

0_SL1, the LE-trees achieved StEq(M

max

and r

) if

F

was applied at a

z#/H-coordinate about 10% lower than in

the combination 0. In the combination 0_SL2 (nos 21 and

26), the corresponding downward shift was 15%.

The critical speed of steady wind

u

ranged between

4 m s

and 57 m s

, depending on the seasonal state of

the tree and whether its crown was loaded with snow

(Table 6).

u

was best described by the social status

of the tree in the stand (Table 1), with greater

u

for

more dominant trees, and then in turn by the tree size

(DBH), by the site (LE or HE), or by the stand density.

When described by the social status,

u

was significantly

higher (Wilcoxon signed-rank test,

P < 0.001) at the

LE-site than at the HE-site. Here, tree no. 23 appears

weak and tree no. 12 strong in relation to their

co-dominant positions. In general, the trees seem to have

adapted their growth, and thus their stem and anchorage

strength and crown size, reasonably well to the expected

wind load. Subject to

u

, the exploitation of the

stem-bending stress capacity, r/r

, was generally highest at

z# ¼ 0.10–0.20H, where the stems also exhibit their

greatest r

(z) (Fig. 3D). The LE-spruces tend to fail in

the stem rather than uproot when exposed to

u

, unlike

the HE-spruces in unfrozen soil.

Table 5. Numerical data related to StEq(M0

maxand rmax), i.e. the simultaneous achievement of M0¼ M0

max and r(z#) ¼ rmax(z#) due to the applied critical force Fcrit, depending on the seasonal tree-state and snow load (cf. Table 4)

IfFcritacts above thez#(Fcrit)/H corresponding to StEq(M0maxand rmax), the stem fails. Otherwise the root–soil system fails (cf. legend in the table). M0, turning moment acting on the root–soil system; M0

max, maximum resistive M0 of the root–soil system; r(z#), stem-bending stress; rmax(z#), stem-bending strength; z#/H, relative coordinate along the stem.

Tree no. Seasonal tree state/snow loada

0 0 FrSt FrSt FrSo FrStSo FrStSo

SL1 SL1 SL1 11 0.85 0.80 0.90 0.90 0.10 0.30 0.25 0.69 0.67 0.81 0.75 0.02 0.09 0.09 12 0.80 0.80 0.85 0.85 0.10 0.20 0.20 0.61 0.61 0.78 0.79 0.04 0.07 0.07 13 0.85 0.80 0.85 0.85 0.15 0.75 0.75 0.74 0.69 0.76 0.77 0.05 0.08 0.33 14 0.80 0.80 0.85 0.85 0.50 0.75 0.70 0.69 0.69 0.75 0.75 0.23 0.62 0.30 15 0.85 0.85 0.90 0.90 0.35 0.80 0.80 0.68 0.67 0.85 0.85 0.07 0.64 0.64 16 0.85 0.85 0.85 0.85 0.70 0.75 0.75 0.78 0.78 0.78 0.78 0.06 0.61 0.61 17 0.80 0.80 0.85 0.85 0.70 0.75 0.75 0.72 0.72 0.77 0.77 0.54 0.65 0.66 21 0.75 0.75 0.53 0.53 22 0.65 0.55 0.30 0.27 23 0.25 0.10 0.06 0.04 24 0.55 0.45 0.17 0.17 25 0.25 0.20 0.05 0.05 26 0.40 0.35 0.11 0.11

a_{Legend for each group of two numbers:}

z#(Fcrit)/H2 StEq(M0maxand rmax);z#/H where r(z#) ¼ rmax(z#).

Table 6. Numerical data related to the critical wind ucrit, depending on tree state and snow load (cf. Table 4)

ucrit causes stem failure if max(r/rmax)¼ 1.00, uprooting if max(r/ rmax) <1.00, and both if the tree is in unstable equilibrium (*). r, stem-bending stress; rmax, stem bending strength (cf. legend in the table). Tree no. Seasonal tree state/snow loada

0 0 FrSt FrSt FrSo FrStSo FrStSo

SL1 SL1 SL1 11 9 4 8 5 21 18 14 0.13 0.69 0.13 0.14 0.13 0.10 0.14 0.71 0.90 0.47 0.50 1.00 1.00 1.00 12 15 8 13 8 33 28 21 0.09 0.10 0.09 0.10 0.09 0.09 0.10 0.76 0.79 0.50 0.52 1.00 1.00 1.00 13 13 8 11 8 32 28 20 0.07 0.08 0.07 0.08 0.07 0.07 0.07 0.57 0.59 0.38 0.38 1.00 1.00 1.00 14 15 13 14 12 38 33 25 0.26 0.27 0.26 0.26 0.26 0.26 0.26 0.42 0.44 0.28 0.28 1.00 0.69 0.71 15 22 16 18 15 46 42 31 0.18 0.18 0.18 0.18 0.07 0.18 0.18 0.53 0.54 0.35 0.35 1.00 0.87 0.89 16 19 16 17 15 45 39 30 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.39 0.39 0.26 0.26 0.97 0.64 0.65 17 32 24 25 20 57 53 41 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.27 0.27 0.18 0.18 0.68 0.45 0.45 21 22 19 0.07 0.07 0.91 0.92 22 21 19 0.09 0.27 0.93 0.96 23 20 11* 0.18 0.96 1.00 1.00 24 34 24 0.19 0.42 1.00 1.00 25 37 24 0.39 0.39 1.00 1.00 26 41 31 0.14 0.23 1.00 1.00

a_{Legend for each group of three numbers: u} crit [m s

1

]; z#/H for max(r/rmax) [–]; max (r/rmax) [–].

Degree of uniform exploitation of the stem-bending capacity, g

All HE- and LE-spruces display a mechanically adapted

stem growth for lateral forces applied slightly above the

area centre of the crown (Fig. 7). Hence, the HE-spruces

exhibit high g-values if

F acts on the upper crown

(0.6 <

z#/H < 0.8), similar to those of the LE-spruces

(0.6 <

z#/H < 0.9). In addition, the stems of the

HE-spruces are well designed for bending due to forces

acting on the lower part of the stem, up to approximately

z#/H¼0.20 (Fig. 7A), especially when loaded with snow

(Fig. 7B). The general g-maximum along the upper stem

is located higher for the LE-spruces than for the

HE-spruces. Physically, this is a result of the LE-trees having

relatively thicker stems along the upper half (Fig. 7D). If

the crown of the HE-spruces is loaded with snow, the

g-maximum is shifted downward (cf. Fig. 7A with B). A

similar downward shift was observed for the LE-spruces.

The mechanically adapted stem growth is only weakly

influenced by the tree states with frozen soil at the HE-site

(Fig. 8), unlike those with snow load. Further, a stem’s

capacity (or need) to adapt its growth mechanically to

different combinations of tree-state and snow load

decreases with the age and size of the tree in terms of

a less differentiated g(

z#). The g(z#)-curves of the

LE-spruces with and without snow load showed a similar

dependency on tree age and size.

Discussion

Model performance and potential sources of error when modelling stem deflections

There are several reasons for the differences between the

parameterized and measured stem flexibility along the

crown section of the stem. First, the branches induce local

variations in grain orientation and stem diameter that

influence the effective flexibility of the stem section.

These variations were not taken into account. Second, the

stem diameter was measured at larger intervals along the

crown than along the stem base. Third, growth properties

(RW and Q) were assessed in only two or three stem discs

along the crown, which is a poor statistical base to predict

the effective secant-modulus of elasticity

E (governed by

Fig. 7. (A–C) Site comparison of degrees of uniform exploitation of bending stress capacity, g, as a function of the relative coordinate along the stemz#/H at which the lateral force F acts: the HE-spruces in the tree-state/snow-load combination 0 and 0_SL1 (A, B) and LE-spruces in the combination 0 (C) (cf. Table 4). The g-maximum along the upper 80% of the tree (dashed lines) is shifted downwards if the crown is loaded with snow (A, B). The dot-dash lines indicate the location of the area centre of the crown. All horizontal lines are site medians. The inclined numbers indicate the tree no. (cf. Table 1 and Table 3). (D) The stem diameter under barkDubnormalized by theDubat 5% stem height, as a function of the relative stem heightz/H. The upper half of the stem is relatively thicker at the LE-site (dashed lines) than at the HE-site (continuous lines).

Equation 1) here. Fourth, the predictive strength of

E is

low for the stem along the crown (Lundstro¨m

et al.,

2008). Bending tests and investigations of the radial

growth of fresh logs from the crown region are required

to analyse and possibly reduce these sources of errors.

However, the overall tree deflection is governed by the

root–soil rotation and the bending of the stem base. The

errors in flexibility along the crown section are therefore

of minor importance when modelling the overall

de-flection of trees.

A geometric and material non-linear analysis can, unlike

a linear analysis, describe

in situ stem deflections

pre-cisely. Since this type of analysis requires non-linear

experimental data, which are not always available, and

more programming than if the analysis is made linear, it

may be of interest to know when such precision is useful.

When estimating critical lateral forces, a strict linear

analysis will always overestimate the critical force (in the

present case by up to 35%), especially if the crown is

loaded with snow (possibly no additional force required

for tree failure). A non-linear approach therefore seems

adequate here as well as for failure analysis in general, for

example when investigating the balance between the stem

and anchorage strengths. As a comparison, the

stream-lining of the crown subject to wind, which is also a

non-linear effect, reduces the effective drag force by up to

75% in strong wind (Equation 4). Unless the crown is

loaded with heavy snow, crown streamlining is therefore

the most important non-linear effect to consider in

analysing tree reactions to wind. It is apparent that the

spruces analysed are normally mechanically favoured by

their overall flexibility, as demonstrated here by a 75%–

35%

¼ 40% higher F

.

Adaptation of stem and anchorage strength to tree states and loads

The strengths of the stem and the anchorage appear, in

general, to be mutually best adapted to the prevailing

combination of seasonal tree state and load, which at both

sites is the wind on the unfrozen tree. The stem and

anchorage strengths were also in balance for the HE-spruces

in frozen soil subject to gliding or creeping snow (Table 5),

especially for the smaller trees. The LE-spruces nos 23 and

25 were two exceptions to this pattern. Their growth seems

to have been more strongly influenced by other criteria

than the balance between the stem and anchorage strengths.

Even so, the overall signs of mechanically adapted stem

and anchorage strengths, which mean an adaptation of tree

growth to acute loading, should be kept in mind when

changes in the forest structure are planned.

The social status of the tree in the stand, the tree size,

the site, and to some extent, the stand density seem, in

turn, to govern the critical values of steady wind,

u

(Table 6). This indicates that the tree adapts, through

appropriate growth allocation, its stem and anchorage

Fig. 8.Comparison of g(z#(F)/H) among three spruces (A–C) of different sizes and cambial ages (atz¼1.3 m) at the HE-site. DBH is the diameter at breast height and the axis symbols are explained in Fig. 7. Each curve corresponds to a particular combination of tree state and snow load (cf. legend in C). The snow load considered is SL1 (cf. Table 4).

strengths as well as its crown size to the wind pressure. In

this study, the steady wind and a uniform wind pressure

on the entire crown were used as a basis for comparing

the

u

of the trees. Evidently, a tree will also experience

(i) gusts due to the wind turbulence (Holbo

et al., 1980),

(ii) dynamic, structural amplification due to tree-wind

resonance (Peltola, 1996; SIA, 2006), (iii) varying types

of wind profile in the canopy (Raupach, 1994; Lo, 1995)

depending on the local topology and stand structure, and

possibly (iv) torsion (Mayer, 1985). When relating

u

to

standard 10-min-wind measurements, the two first effects

tend to counterbalance the third. It is, however, obvious

that predictions of wind-induced tree failure will require

additional and more specific measurements than those

made in the present study. Nevertheless, this analysis

clearly shows that the studied trees in frozen soil will

never fail due to wind load alone, and that the trees with

unfrozen stem and soil and snow-loaded crowns are at

greatest risk of wind failure. The HE-trees are accustomed

to frost (note that light frost is more frequent than ‘Fr’ in

Table 4) and appear to allocate growth to the anchorage

and the stem in accordance. In fact, they seem to rely on

their anchorage and stem strengthening when the crown is

loaded with snow, which is another sign of site-specific

mechanical growth adaptation.

Field observations at the HE-site over the last few

decades (H Hefti, municipal forester at the HE site,

personal communication) reveal only a few tree failures.

These were mainly due to local Foehn-storms, small slabs

of gliding snow, and rare tree-top failure due to

snow-loaded crowns. In comparison, more frequent tree failures

due to winter storms were observed at the LE-site, of

which about one-third were stem and two-thirds root–soil

failure (B Blo¨chliger, municipal forester at the LE-site,

personal communication). These observations are in line

with the model simulations in this study.

Degree of uniform exploitation of the stem-bending capacity, g

Stem thigmomorphogenesis reflects the local prevailing

seasonal combination of tree state and load. In fact, the

degree of uniform exploitation of bending stress capacity

g(z#) (Equation 8) shows that both the taper and the material

properties of the stem mechanically adapt to frequent

complex loading, independent of the growth situation. This

is especially true for young trees (Fig. 8). The general

maxima of g(

z#) of about 0.75 (Fig. 7) shows that the stem

adapts its growth to the effectively induced wind load, just

like the strengths of the stem and of the root–soil system.

Actually, the effective height of equivalent wind-force

application

z#(F) on the crown could be estimated to

20–25% above the crown’s area centre if the crown’s area

was considered by segments of height in Equation 7 and an

approximate mean wind profile in the canopy was assumed

(on the basis of Zoumakis, 1993; Raupach, 1994).

In terms of stem thigmomorphogenesis, the thicker stems

of the LE-spruces along the crown section of the stem, and

thus the slightly higher located g(

z#)-maxima for the

LE-compared to the HE-spruces, could be a result of

differ-ences in the wind pressure profiles, with relatively higher

pressures close to the tree tops at the LE-site. Concerning

the observed response in stem growth to the prevailing load

combination, the snow load SL1 is an exception. Although

the tree crown is mostly not covered with snow, the stem

seems to ‘remember’ this critical additional load and adapts

its growth accordingly. The downward shift in general

g(z#)-maxima for trees with snow-loaded crowns seems

natural, since the equivalent wind force acting on the crown

is likely to shift downward if the branches are loaded with

snow. This interpretation is supported by the fact that the

application height of

F

at StEq(M

and r

) was also

shifted downward if the crown was loaded with snow.

It is known that trees also exhibit stem failure due to

shearing (Mattheck and Breloer, 1994), but no signs of

stem thigmomorphogenesis due to shear stresses were

found. To ascertain this will require a shear-stress

model-ling with more detail than could be included in this study.

Nevertheless, the formulation of the degree of uniform

exploitation of a tree’s stem-bending stress capacity

pre-sented here provides further strong evidence that tree stems

are able to adapt their growth mechanically.

This study covers numerous tree and climate

parame-ters, with an emphasis on drawing general conclusions

rather than doing a detailed analysis of the spatiotemporal

growth response of spruce to mechanical stress. For more

insight into the thigmomorphogenesis of Norway spruce,

it would be worthwhile analysing how specific growth

processes develop in detail under the influence of variable

mechanical stress and climatic conditions.

Acknowledgements

We thank Silvia Dingwall for revising the text and the anonymous reviewer for constructive comments.

Appendix

List of symbols and notations and their definitions and units used in the study

Notation Description Unit

A Area of the stem cross-section m2

B Bark thickness m, mm

D, Dub;x, y, 05

Stem diameter over and under bark; reference tox-direction, y-direction, and to 5% tree height (cf. Table 2)

m, mm

DBH Stem diameter over bark at breast height (z¼1.3 m)

m, cm E Secant-modulus of stem bending elasticity

¼ r/e

GPa e Stem-bending strain (without the contribution

from shear deformation)

–

References

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Notation Description Unit

F ; Finit;Fcrit

Force applied on the tree in thex-direction; F initially applied; critical F, causing stem-bending failure (r¼rmax), or tree uprooting (M0_{ M}0

max), or both (M0maxand rmax) simultaneously

N

/ Rotation around they-axis of the root-soil system

rad,
G Secant-modulus of shearing elasticity along

the stem

MPa

c Stem-shear strain along the stem –

H Tree height m

g Degree of uniform exploitation of the stem-bending capacity (cf. Equation 8)

– HE, LE High-elevation site, low-elevation site

M Turning moment around they-axis of the stem section

MPa M0;M0

max Turning moment around they-axis of the

root–soil system; maximum resistiveM0 MPa MOE Secant-modulus of stem bending elasticity

defined at r¼0.4rmax

GPa

N Normal force (along the stem) N

Q Knottiness (relative frequency of knots) in the stem cross-section

– r Radial coordinate of the stem, ranging from

the stem centre to the bark (D0/2)

m RW;i,o Width of annual rings; reference to the

mean of the inner 75% radial part of the stem cross-section and to the mean of the remaining outer part

mm

r, rmax Bending stress and strength of the stem cross-section

MPa SL1, SL2 Snow load on the crown m
1height (1¼

moderate; 2¼heavy, cf. Table 4 and the text)

kg m
1 StEq Static equilibrium for the tree model when

subjected toFcrit(cf. explanation ofFcritabove) SWE Snow-water equivalent precipitation relating

to the amount of snow

mm s, smax Shear stress and strength of the stem section

in the direction along the stem

MPa u; ucrit Wind speed; criticalu causing stem-bending

failure (r¼rmax) or tree uprooting (M0 M0max)

m s
1

V Shear force (across the stem) N

x, y, z Global (Cartesian) coordinates of the tree: origin at stem base;x¼horizontal stem deflection; z¼height above origin (cf. Fig. 1)

m

x#, z# Local coordinates of the deflecting stem; x#¼sideway deflection of each stem element; z#¼coordinate along the deflecting stem, ranging from 0 toH (cf. Fig. 1).

m

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